Consider two time series, $\{X(t),Y(t)\in \Re |t\geq0\}$, whose correlation is 1. That is, $$ \rho_{X,Y}=\frac{\mathbb{E}\left[\left(X - \mathbb{E}[X]\right)\left(Y-\mathbb{E}[Y]\right)\right]}{\sqrt{\mathbb{V}[X]\mathbb{V}[Y]}} = 1. $$ Now consider each series' respective z-score, defined (for $X(t)$) as $$ z_{X}(t) = \frac{X(t) - \mathbb{E}[X]}{\sqrt{\mathbb{V}[X]}}. $$ Is there any way to show that, because $\rho_{X,Y} = 1$, $z_{X}(t) = z_{Y}(t) \;\forall t$?